2.1 Nonlinear loads

Concerning the definition of a nonlinear load, it is necessary to specify linearity. Linearity is a characteristic used to describe linear loads, and it corresponds to a property in which loads exclusively produce fundamental sinusoidal current if supplied by a sinusoidal voltage source at fundamental frequency [16]. In contrast, nonlinear loads provide distorted current waveforms, thus injecting harmonic components in the system [17]. Load harmonics higher than fundamental frequency are commonly represented with a resistance-inductance-capacitance (also known as RLC) circuit in parallel with a current source, as shown in Figure 1.

Nonlinear loads act as sources of harmonic currents whose frequencies are multiple of the fundamental frequency. Harmonics circulated from the load to the source and, depending on the topology of the network, harmonic current can

By expressing (4) in its phasor form, the result is as given in (5).

q

describe the total harmonic distortion (THD) [20], which is a frequently used

THDI <sup>¼</sup> IH

Another simple way to describe the harmonic influence over the fundamental frequency sinusoid is the distortion factor γ. Its formulation is as follows [21]

> I I 2 <sup>1</sup>rms <sup>q</sup> <sup>¼</sup> <sup>1</sup> ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

> > I1rms

By replacing (8) in (9) and solving for IH, the relationship between the magni-

<sup>1</sup> <sup>þ</sup> THD<sup>2</sup> I

vector configuration approach (applied for nonlinear loads) as presented in [22]. Then, the phasor in (5) can be written in component form as follows [17]:

<sup>e</sup><sup>I</sup> <sup>¼</sup> IA^<sup>i</sup> <sup>þ</sup> IX^<sup>j</sup> <sup>þ</sup> IH^

I 2 <sup>A</sup> þ I 2 <sup>X</sup> þ I 2 H

s

q

I ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I 2 <sup>A</sup> þ I 2 X <sup>1</sup> � <sup>H</sup><sup>2</sup>

With respect to the vector representation of the phasor given in (2), the domains ^<sup>i</sup> and ^<sup>j</sup> correspond to the active ð Þ IA and reactive ð Þ IX current components, respec-

I1rms Irms

<sup>¼</sup> IH Irms

I1rms

<sup>¼</sup> <sup>I</sup>1rms ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I 2 <sup>1</sup>rms þ I 2 H

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I 2 <sup>1</sup>rms þ I 2 H

and Irms ¼

measure of harmonic levels. Mathematically, it is expressed as [21]:

<sup>γ</sup> <sup>¼</sup> <sup>I</sup>1rms Irms

<sup>γ</sup> <sup>¼</sup> <sup>I</sup>1rms ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>1</sup>rms <sup>þ</sup> THD<sup>2</sup>

THDI<sup>γ</sup> <sup>¼</sup> IH

tude of the harmonic part and the total current magnitude is obtained:

tively. Nevertheless, in the presence of nonlinear loads, the harmonic

I ¼

IH <sup>¼</sup> THDI ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

I 2

Later, by multiplying (6) and (7), the expression is:

Solving for IH in (6) and replacing it in (7):

where H is denoted the harmonic factor.

componentð Þ IH appears in a third domain ^

Then the apparent current magnitude is

By replacing (10) into (12) and solving for I,

123

where IH ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑NH h¼2 I 2 h 2

DOI: http://dx.doi.org/10.5772/intechopen.84345

Solid-State Transformer for Energy Efficiency Enhancement

q

eI ¼ I1rms∡θ<sup>1</sup> þ IH∡θ<sup>h</sup> ¼ Irms∡θ<sup>I</sup> (5)

. This last formulation is useful to

<sup>q</sup> (7)

<sup>q</sup> (8)

<sup>1</sup> <sup>þ</sup> THD<sup>2</sup> I

<sup>q</sup> <sup>I</sup> <sup>¼</sup> HI (10)

k. This fact is attributed to the power

k (11)

(6)

(9)

(12)

(13)

Figure 1. Equivalent circuit of a nonlinear load.

Figure 2. The effect of harmonics.

spread to other loads. These distorted current components may cause voltage spikes and terrible damage to nearby equipment. Note these phenomena in Figure 2.

The fundamental current waveform as a function of time t can be represented as in (1), where the term I<sup>1</sup> represents the current fundamental peak amplitude, ω<sup>0</sup> is the fundamental angular frequency, and θ<sup>1</sup> is the phase angle.

$$i\_1(t) = I\_1 \cos\left(\alpha\_0 t + \theta\_1\right) \tag{1}$$

For simplicity, it is common to represent a sinusoidal function in its phasor form, where it is written as a complex number with amplitude and phase. The phasor amplitude is obtained from the root mean square value of the fundamental sinusoid function amplitude, that is, <sup>I</sup>1rms <sup>¼</sup> <sup>I</sup>1<sup>=</sup> ffiffi 2 <sup>p</sup> [18]. Applying this criterion to (1), its phasor representation is as follows:

$$
\ddot{I}\_1 = I\_{1rms} \mathfrak{A} \theta\_1 \tag{2}
$$

In the presence of harmonics, waves are distorted and become a function of the total number of harmonics NH. Considering this fact and using Fourier series, the current has the form as presented in (3) [19].

$$\dot{x}(t) = \sum\_{h=1}^{NH} I\_h \cos\left(h\alpha\_0 t + \theta\_h\right) = I\_1 \cos\left(\alpha\_0 t + \theta\_1\right) + \sum\_{h=2}^{NH} I\_h \cos\left(h\alpha\_0 t + \theta\_h\right) \tag{3}$$

Notice that the left-hand side term of the sum is the fundamental frequency sinusoid, which has exactly the form as presented in (1), while the right-hand side term is the harmonic currentih, i.e., the distortion wave. Then, (3) can be rewritten as:

$$\dot{\mathbf{u}}(t) = \dot{\mathbf{r}}(t)^{\prime} + \dot{\mathbf{r}}\_{h}(t) \tag{4}$$

By expressing (4) in its phasor form, the result is as given in (5).

$$
\ddot{I} = I\_{1rms} \mathfrak{A} \theta\_1 + I\_H \mathfrak{A} \theta\_h = I\_{rms} \mathfrak{A} \theta\_I \tag{5}
$$

where IH ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ∑NH h¼2 I 2 h 2 q and Irms ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I 2 <sup>1</sup>rms þ I 2 H q . This last formulation is useful to describe the total harmonic distortion (THD) [20], which is a frequently used measure of harmonic levels. Mathematically, it is expressed as [21]:

$$\text{THD}\_I = \frac{I\_H}{I\_{1rms}} \tag{6}$$

Another simple way to describe the harmonic influence over the fundamental frequency sinusoid is the distortion factor γ. Its formulation is as follows [21]

$$\chi = \frac{I\_{1rms}}{I\_{rms}} = \frac{I\_{1rms}}{\sqrt{I\_{1rms}^2 + I\_H^2}} \tag{7}$$

Solving for IH in (6) and replacing it in (7):

$$\chi = \frac{I\_{1rms}}{\sqrt{I\_{1rms}^2 + THD\_I^2 I\_{1rms}^2}} = \frac{1}{\sqrt{1 + THD\_I^2}}\tag{8}$$

Later, by multiplying (6) and (7), the expression is:

$$\text{THD}\_{I\bar{I}}\text{y} = \frac{I\_H}{I\_{1rms}} \frac{I\_{1rms}}{I\_{rms}} = \frac{I\_H}{I\_{rms}} \tag{9}$$

By replacing (8) in (9) and solving for IH, the relationship between the magnitude of the harmonic part and the total current magnitude is obtained:

$$I\_H = \frac{\text{THD}\_I}{\sqrt{\mathbf{1} + \text{THD}\_I^2}} I = \text{HI} \tag{10}$$

where H is denoted the harmonic factor.

With respect to the vector representation of the phasor given in (2), the domains ^<sup>i</sup> and ^<sup>j</sup> correspond to the active ð Þ IA and reactive ð Þ IX current components, respectively. Nevertheless, in the presence of nonlinear loads, the harmonic componentð Þ IH appears in a third domain ^ k. This fact is attributed to the power vector configuration approach (applied for nonlinear loads) as presented in [22]. Then, the phasor in (5) can be written in component form as follows [17]:

$$
\widetilde{I} = I\_A \hat{\mathbf{i}} + I\_X \hat{\mathbf{j}} + I\_H \hat{\mathbf{k}} \tag{11}
$$

Then the apparent current magnitude is

$$I = \sqrt{I\_A^2 + I\_X^2 + I\_H^2} \tag{12}$$

By replacing (10) into (12) and solving for I,

$$I = \sqrt{\frac{I\_A^2 + I\_X^2}{1 - H^2}}\tag{13}$$

spread to other loads. These distorted current components may cause voltage spikes and terrible damage to nearby equipment. Note these phenomena in Figure 2.

For simplicity, it is common to represent a sinusoidal function in its phasor form, where it is written as a complex number with amplitude and phase. The phasor amplitude is obtained from the root mean square value of the fundamental

In the presence of harmonics, waves are distorted and become a function of the total number of harmonics NH. Considering this fact and using Fourier series, the

Notice that the left-hand side term of the sum is the fundamental frequency sinusoid, which has exactly the form as presented in (1), while the right-hand side term is the harmonic currentih, i.e., the distortion wave. Then, (3) can be rewritten

i tðÞ¼ i tð Þ0

I

Ih cosð Þ¼ hω0t þ θ<sup>h</sup> I<sup>1</sup> cosð Þþ ω0t þ θ<sup>1</sup> ∑

the fundamental angular frequency, and θ<sup>1</sup> is the phase angle.

sinusoid function amplitude, that is, <sup>I</sup>1rms <sup>¼</sup> <sup>I</sup>1<sup>=</sup> ffiffi

(1), its phasor representation is as follows:

current has the form as presented in (3) [19].

i tðÞ¼ ∑ NH h¼1

Figure 1.

Figure 2.

The effect of harmonics.

Equivalent circuit of a nonlinear load.

Research Trends and Challenges in Smart Grids

as:

122

The fundamental current waveform as a function of time t can be represented as in (1), where the term I<sup>1</sup> represents the current fundamental peak amplitude, ω<sup>0</sup> is

i1ðÞ¼ t I<sup>1</sup> cosð Þ ω0t þ θ<sup>1</sup> (1)

e<sup>1</sup> ¼ I1rms∡θ<sup>1</sup> (2)

NH h¼2

<sup>p</sup> [18]. Applying this criterion to

Ih cosð Þ hω0t þ θ<sup>h</sup> (3)

þ ihð Þt (4)

2

The last formulation describes in functional way the mathematical model for a nonlinear load.

2.3 Solid-state transformer

Solid-State Transformer for Energy Efficiency Enhancement

DOI: http://dx.doi.org/10.5772/intechopen.84345

these requirements are shown in Table 3.

Requirements Description

energies

Storage systems

quality

Remote operation

supply

Various functional requirements for the SST.

Multilevel cascade rectifier Cuk converter

Typical power electronics topology for the SST stages.

Integration Renewable

Coordination Power

Consumption Power

Table 3.

Table 4.

125

The SST allows isolation between medium- and low-AC voltage sides as any conventional transformer. Additionally, it allows the isolation and clearance of faulty conditions from both sides, as well as anomalies encountered in the AC or DC sides. Its DC link is highly attractive for the integration of photovoltaic energy, storage systems with uninterrupted power supply devices, or even future local DC grids. In order to accomplish all these features, its topology has several stages of power electronic blocks depending on the functionalities required. Thus, the SST can be designed depending on the type of application [23]. As a key technology in the implementation of the smart grid, its topology will heavily depend on the end user consumption and the integration and coordination features required. Some of

As the modular arrangement of the SST depends on the grid requirements, several topologies have been proposed in the literature. Generally, the energy can be processed in three main stages: rectification, the same level AC-AC or DC-DC conversion, and inversion. Some of the available solutions to these stages are shown in Table 4. To provide a wider classification system for the SST, the level of modularity can be determined with respect to power flow direction, connection to

panels) or LVAC (e.g., wind micro-turbines)

Voltage magnitude (e.g., power factor correction)

Integration of distributed generation on LVDC (e.g., photovoltaic

Integration of energy storage system (e.g., battery systems) or devices

Reactive compensation (e.g., fast response to voltage disturbances due to

Communication functionalities to be integrated to higher management systems (e.g., SCADA systems, energy management systems (EMS), outage management systems (OTS), wide area management systems (WAMS), and other early awareness systems with synchro-phasor

Several voltage level requirements: HVAC to LVAC, HVAC to LVAC +

End user consumption such as LVAC loads (linear and nonlinear loads)

Voltage unbalance (e.g., rapid response to sags, swells, and all the harmonics originated at the load or perceived at the device's input) Other quality events (e.g., electromagnetic transients, frequency

three-phase systems, and connection to the medium-voltage level [24].

with UPS functionality

variations)

capabilities)

LVDC, etc.

and LVDC loads

Active front-end rectifiers Bi-directional DC-DC dual-active-bridge converter

Rectification Same level DC-DC conversion Inversion

Full-bridge rectifiers Buck/boost/buck-boost converter Full-bridge inverters

reactive energy unbalances)
